Classifying relative equilibria. II

Abstract

We announce several theorems which suggest a minimal classification of relative equilibria in the planar «-body problem. These theorems also answer several questions on the nature of degenerate relative equilibria classes which were asked recently by S. Smale [3]. A summary of previous results can be found in an earlier paper [1]. It is a pleasure to thank S. Smale for encouragement in this work.1. Morse theory and relative equilibria. We study the critical set of a real analytic function V m < 0 on a real analytic manifold X m where n > 3 and m = (m 1 , . . . , m n ) GR"_ are fixed. Critical points of V m correspond in a 1-1 fashion to classes of relative equilibria. V m always has a compact critical set which we may investigate by Morse theory even when degenerate critical points exist [2].The integral singular homology of X m (a manifold which is homeomorphic to a Stein manifold P n _ 2 (Q ~ A w _ 2 ) is given by a recurrence relation [1]. This suggests that there is a uniform lower bound on the number of critical points of each index of V m which is given by recurrence. As a first step toward classifying relative equilibria Theorem 1 gives such a relation.In Theorem 2 we assert that V m is a Morse function for any n > 3 and for almost all m G R^_ (in the sense of Lebesgue measure).Theorem 3 answers the question: Is V m always a Morse function? Finally, we examine the case of four masses to show how a degeneracy of V m arises. An interpretation of Theorem 1 in the degenerate case sheds light on the creation and annihilation of relative equilibria.2. Main theorems. In this paragraph for any /, 0 < / < In -4, let M/00 denote a uniform lower bound to the number of critical points of V m AMS (MOS) subject classifications (1970). Primary 70F10; Secondary 57D70.

Finiteness of central configurations of five bodies in the plane

“…In 1975, Palmore [25] gave a simple example of a degenerate central configuration, an equilateral triangle of bodies with unit mass and a fourth body with mass (64 √ 3+81)/249 at the center of the triangle. Simó [28] showed how the number of 4-body central configurations with a given body in the interior of the triangle formed by the other bodies varies when the four masses vary.…”

Bifurcation of Relative Equilibria of the (1+3)-Body Problem

“…Thus there are exactly N !/2 classes of collinear central configurations for a given set of N positive masses; see Moulton [32]. Using Morse theory Palmore obtained a lower bound of the number of central configurations under a nondegeneracy assumption [33]. For N = 4, there are 12 collinear central configurations, and Palmore's lower bound is 34.…”

A note on the Dziobek central configurations

“…For a classical background, see the sections on central configurations in the books of Wintner [17] and Hagihara [6]. For a modern background see, for instance, the papers of Albouy and Chenciner [2], Albouy and Kaloshin [3], Hampton and Moeckel [7], Moeckel [9], Palmore [13], Saari [14], Schmidt [15], Xia [18], ... One of the reasons why central configurations are important is that they allow to obtain the unique explicit solutions in function of the time of the n-body problem known until now, the homographic solutions for which the ratios of the mutual distances between the bodies remain constant. They are also important because the total collision or the total parabolic escape at infinity in the n-body problem is asymptotic to central configurations, see for more details Dziobek [5] and [14].…”